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__Approximate solutions of fractional differential equations with Riesz fractional derivatives in a finite domain__

Y. Takeuchi and R. Suda

**Abstract**

This paper states approximate solutions of fractional differential equations with Riesz fractional derivatives (FDE-RFD) in a finite domain. FDE-RFD are used in many fields, especially in the prediction of the diffusion of radioactive materials [1]. FDE-RFD in an infinite domain have already solved by using fractional Fourier transform [2]. In contrast, we consider how to solve FDE-RFD in a finite domain.
In this paper, we assume FDE-RFD in a finite domain can be solved by the method of separation of variables and the analytical solution is constructed by polynomials. As a result, we find the analytical solution can be obtained by solving a linear equation with infinite size, and we get approximate solutions from solving a linear equation which is approximated to finite size. Note that approximate solutions in this paper mean the solutions which are closely related to asymptotic expansions.

**Bibliography**

[1]
A.S. Chaves, A fractional diffusion equation to describe Levy flights, Physics Letters A 239(1998).

[2]
E. Sousa, Finite difference approximations for a fractional advection diffusion problem, Journal of Computational Physics 228(2009).